Pay Periods and Current Periods

this should do

PayPeriod = 
-- variable declaration section
VAR __CurrentDate = 'Calendar'[Date]    -- returns the date for current row (because it's calculated column)
-- output section
RETURN
IF(
    DAY(__CurrentDate) <= 15,    -- checks if the day of current row date is <= 15
    __CurrentDate - DAY(__CurrentDate) + 15 ,    -- if it is, it takes the date subtracts the number of days of this date (which gives 0) and adds 15
    EOMONTH(__CurrentDate, 0)    -- if it's not it gives the last day of the month
    )

Period =
-- variable declaration section
VAR __CurrentPayPeriod = 'Calendar'[PayPeriod]    -- returns the pay period for a current row
VAR __Today = TODAY ()    -- returns the today's date
VAR __TodayPayPeriod =    -- returns the pay period for today (using method from [PayPeriod]
    IF (
        DAY ( __Today ) <= 15,
        TODAY () + 15 - DAY ( TODAY () ),
        EOMONTH ( TODAY (), 0 )
    )
VAR __PreviousPayPeriod =    -- calculates the maximum date in PayPeriod column that is strictly before today's pay period
    CALCULATE (
        MAX ( 'Calendar'[PayPeriod] ),
        FILTER ( ALL ( 'Calendar' ), 'Calendar'[PayPeriod] < __TodayPayPeriod )
    )
VAR __FuturePayPeriod =    -- calculates the minimum date in PayPeriod column that is strictly after today's pay period
    CALCULATE (
        MIN ( 'Calendar'[PayPeriod] ),
        FILTER ( ALL ( 'Calendar' ), 'Calendar'[PayPeriod] > __TodayPayPeriod )
    )
-- output section
RETURN
    IF (
        __TodayPayPeriod = __CurrentPayPeriod,
        "Current Period",
        IF (
            __CurrentPayPeriod = __PreviousPayPeriod,
            "Previous Period",
            IF ( __CurrentPayPeriod = __FuturePayPeriod, "Future Period", "Other" )
        )
    )

I've added the comments in the accepted solution, lat me know if something is not clear. I use variables, they are explained in more detail here
https://www.sqlbi.com/articles/variables-in-dax/

also you can see what is calculated in each step by changing, what's returned, e.g. code below will return the today pay period, you can only return one thing at a time, I commented the actual answer in the code below)

Period =
-- variable declaration section
VAR __CurrentPayPeriod = 'Calendar'[PayPeriod] -- returns the pay period for a current row
VAR __Today =
    TODAY () -- returns the today's date
VAR __TodayPayPeriod =
    -- returns the pay period for today (using method from [PayPeriod]
    IF (
        DAY ( __Today ) <= 15,
        TODAY () + 15
            - DAY ( TODAY () ),
        EOMONTH ( TODAY (), 0 )
    )
VAR __PreviousPayPeriod =
    -- calculates the maximum date in PayPeriod column that is strictly before today's pay period
    CALCULATE (
        MAX ( 'Calendar'[PayPeriod] ),
        FILTER ( ALL ( 'Calendar' ), 'Calendar'[PayPeriod] < __TodayPayPeriod )
    )
VAR __FuturePayPeriod =
    -- calculates the minimum date in PayPeriod column that is strictly after today's pay period
    CALCULATE (
        MIN ( 'Calendar'[PayPeriod] ),
        FILTER ( ALL ( 'Calendar' ), 'Calendar'[PayPeriod] > __TodayPayPeriod )
    ) -- output section
RETURN
    __TodayPayPeriod
/*
    IF (
        __TodayPayPeriod = __CurrentPayPeriod,
        "Current Period",
        IF (
            __CurrentPayPeriod = __PreviousPayPeriod,
            "Previous Period",
            IF ( __CurrentPayPeriod = __FuturePayPeriod, "Future Period", "Other" )
        )
    )
*/